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  • Let be a matrix of order <img alt="\mathrm{2\times 2}" src="https://entrancecorner.oncodecogs.com/gif

Let \mathrm{A} be a matrix of order \mathrm{2\times 2} whose entries are from the set \mathrm{\{0,1,2,3,4,5\}}.If the sum of all the entries of \mathrm{A} is a prime number \mathrm{p, 2<p<8}, then the number of such matrices \mathrm{A} is _______.

Option: 1

180


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{A=\left[\begin{array}{cc} a & b \\ c & d \end{array}\right], \quad b=3,5,7} \\

\mathrm{b=3 \Rightarrow a+b+c+d=3, \quad a, b, c, d \in\{0,1,2,3\}} \\

\mathrm{\text { So } \quad n_{1}= ^{3+4-1}c_{4-1}={ }^{6} c_{3}=20}

\mathrm{p=5 \Rightarrow a+b+c+d=5, \quad a, b, c, d \in\{0,1,2,3,4,5\}} \\

\mathrm{\Rightarrow n_{2}=^{5+4-1} c_{4-1}={ }^{8} C_{3}=56 .}

\mathrm{p=7 \Rightarrow a+b+c+d=7, \quad a, b, c, d \in\{0,1,2,3,4,5\}} \\

\mathrm{n_{3} =^{7+4-1} C_{4-1}-(6,1,0,0)-(7,0,0,0)} \\

\mathrm{=^{10} c_{3}-\frac{4 !}{2 !}-\frac{4 !}{3} !} \\

\mathrm{=120-12-4=104} \\

\mathrm{Total=n_{1}+n_{2}+n_{3}=20+56+104=180 }

Posted by

Ritika Jonwal

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